針對有限元素分析而言，六面體網格化所得到的分析結果與收斂性比起其他網格方式來得更好，所以近年來有關六面體自動網格化的研究越來越多，但不管何種演算法都尚未穩健，因此本文綜觀六面體網格產生的演算法歸納整理出三大類：直接法、間接法、空間分割法，並一一詳述其性質。
　　由於四邊形網格是六面體網格的基礎，所以本文對四邊形網格產生的方式也有介紹，而且四邊形網格也可歸納整理出三大類：直接法、間接法、空間分割法。
　　除此之外，本文也提出利用中間軸定理分割模型並搭配Quad-Morph的方式產生四邊形網格，針對六面體網格而言，本文提出利用中間面搭配Mapping的方式產生六面體網格。

　　For finite element analysis, hexahedral element mesh can make the computation more efficient and thus in general is a better solution. Therefore, there are more and more researches about automatic hexahedral mesh generation in these years. But the algorithms about automatic hexahedral mesh generation are not robust. This thesis reviews algorithms about hexahedral mesh generation and classifies these algorithms into three categories: direct, indirect, and spatial decomposition.

　　Because the quadrilaterals are the basis for hexahedra mesh generation, it will also be introduced in this study. In addition, this thesis will use the medial object to divide the model into simple geometry and then utilizes the Quad-morph algorithm to construct the quadrilaterals and generates the hexahedra with mapping method.

[1] Blacker, Ted D. and Michael B. Stephenson, “Paving: A New Approach to Automated Quadrilateral Mesh Generation,” International Journal for Numerical Methods in Engineering, Vol.32, pp.811-847 (1991).
[2] Lober, Randy R., Timothy J. Tautges, and Rich A. Cairncross, “The Parallelization of an Advancing-Front, All-Quadrilateral Meshing Algorithm for Adaptive Analysis,” Proceedings 4th International Meshing Roundtable, pp.59-70 (1995).
[3] Cass, Roger J., Steven E. Benzley, Ray J. Meyers, and Ted D. Blacker, “Generalized 3-D Paving: An Automated Quadrilateral Surface Mesh Generation Algorithm,” International Journal for Numerical Methods in Engineering, Vol.39, pp.1475-1489 (1996).
[4] White, David R. and Paul Kinney, “Redesign of the Paving Algorithm: Robustness Enhancements through Element by Element Meshing,” Proceedings 6th International Meshing Roundtable, Sandia National Laboratories, pp.323-335 (1997).
[5] Zhu, J. Z., O. C. Zienkiewicz, E. Hinton, and J. Wu, “A New Approach to the Development of Automatic Quadrilateral Mesh Generation,” International Journal for Numerical Methods in Engineering, Vol.32, pp.849-886 (1991).
[6] Owen, Steven J., Matthew L. Staten, Scott A. Canann, and Sunil Saigal, “Advancing Front Quadrilateral Meshing Using Triangle Transformations,” Proceedings 7th International Meshing Roundtable, pp.409-428 (1999).
[7] Baehmann, Peggy L., Scott L. Wittchen, Mark S. Shephard, Kurt R. Grice, and Mark A. Yerry, “Robust Geometrically-based, Automatic Two-Dimensional Mesh Generation,” International Journal for Numerical Methods in Engineering, Vol.24, pp.1043-1078 (1987).
[8] Shimada, K., J. -H. Liao, and T. Itoh, “Quadrilateral Meshing with Directionality Control through the Packing of Square Cells,” Proceedings 7th International Meshing Roundtable, (1998).
[9] Nowottny, Dietrich, “Quadrilateral Mesh Generation via Geometrically Optimized Domain Decomposition,” Proceedings 6th International Meshing Roundtable, pp.309-320 (1997).
[10] Chae, Soo-Won and Jung-Hwan Jeong, “Unstructured Surface Meshing Using Operators,” Proceedings 6th International Meshing Roundtable, pp.281-291 (1997).
[11] Joe, Barry, “Quadrilateral Mesh Generation in Polygonal Regions,” Computer Aided Design, Vol.27, pp.209-222 (1995).
[12] Tam, T. K. H. and C. G. Armstrong, “2D Finite Element Mesh Generation by Medial Axis Subdivision,” Advances in Engineering Software, Vol.13, pp.313-324 (1991).
[13] Quadros, W. R., K. Ramaswami, F. B. Prinz, and B. Gurumoorthy, “LayTracks: A New Approach to Automated Quadrilateral Mesh Generation using MAT,” Proceedings 9th International Meshing Roundtable, pp.239-250 (2000).
[14] White, David R., Lai Mingwu, Steven E. Benzley, and Gregory D. Sajaardema, “Automated Hexahedral Mesh Generation by Virtual Decomposition,” Proceedings 4th International Meshing Roundtable, pp.165-176 (1995).
[15] Blacker, Ted D., “The Cooper Tool,” Proceedings 5th International Meshing Roundtable, pp.13-30 (1996).
[16] Shih, Bih-Yaw and Hiroshi Sakurai, “Automated Hexahedral Mesh Generation by Swept Volume Decomposition and Recomposition,” Proceedings 5th International Meshing Roundtable, pp.273-280 (1996).
[17] Chiba, N., I. Nishigaki, Y. Yamashita, C. Takizawa, and K. Fujishiro, “An Automatic Hexahedral Mesh Generation System Based on the Shape-Recognition and Boundary-Fit Methods,” Proceedings 5th International Meshing Roundtable, pp.281-290 (1996).
[18] Liu, Shang-Sheng and Rajit Gadh, “Basic Logical Bulk Shapes (BLOBs) for Finite Element Hexahedral Mesh Generation,” Proceedings 5th International Meshing Roundtable, pp.291-304 (1996).
[19] Liu, Shang-Sheng and Rajit Gadh, “Automatic Hexahedral Mesh Generation by Recursive Convex and Swept Volume Decomposition,” Proceedings 6th International Meshing Roundtable, pp.217-231 (1997).
[20] Staten, Matthew L., Scot A. Canann, and Steven J. Owen, “BMSweep: Locating interior Nodes During Sweeping,” Proceedings 7th International Meshing Roundtable, pp.7-18 (1998).
[21] Knupp, Patrick M., “Next-Generation Sweep Tool: A Method For Generation All-Hex Meshes on Two-And-One-Half Dimenstional Geomtries,” Proceedings 7th International Meshing Roundtable, pp.505-513 (1998).
[22] Shepherd, Janson, Scott A. Mitchell, Patrick Knupp, and David White, “Methods for Multisweep Automation,” Proceedings 9th International Meshing Roundtable, pp.77-87 (2000).
[23] Schneiders, R., “Automatic Generation of Hexahedral Finite Element Meshes,” Proceedings 4th International Meshing Roundtable, pp.103-114 (1995).
[24] Schneiders, R., R. Schindler, and F. Weiler, “Otree-based Generation of Hexahedral Element Meshes,” Proceedings 5th International Meshing Roundtable, (1996).
[25] Schneiders, R., “An Algorithm for the Generation of Hexahedral Element Meshes Based on An Octree Technique,” Proceedings 6th International Meshing Roundtable, pp.195-196 (1997).
[26] Walton, Kirk S., Steven E. Benzly, and Janson Shepherd, “Sculpting: An Improved Inside Out Scheme for All Hexahedral Meshing,” Proceedings 11th International Meshing Roundtable, pp.153-160 (2002).
[27] Owen, Steven J., “Constrained Triangulation: Application to Hex-Dominant Mesh Generation,” Proceedings 8th International Meshing Roundtable, pp.31-41 (1999).
[28] Canann, Scott A., “Plastering and Optismoothing: New Approach to Automated, 3D Hexahedral Mesh Generation and Mesh Smoothing,” Ph. D. Dissertation, Brigham Young University, Provo, UT. (1991).
[29] Blacker, Ted D. and R. J. Myers, “Seams and Wedges in Plastering: A 3D Hexahedral Mesh Generation Algorithm,” Engineering With Computers, Vol.2, pp.83-93 (1993).
[30] Tautges, Timothy J. and Scott A. Mitchell, “Whisker Weaving: Invalid Connectivity Resolution and Primal Construction Algorithm,” Proceedings 4th International Meshing Roundtable, pp.115-127 (1995).
[31] Mitchell, Scott A. and Timothy J. Tautges, “Pillowing Doublets: Refining A Mesh to Ensure that Faces Share at Most One Edge,” Proceedings 4th International Meshing Roundtable, pp.231-240 (1995).
[32] Tautges, Timothy J., Ted D. Blacker, and Scott A. Mitchell, “The Whisker Weaving Algorithm: A Connectivity-Based Method for All-Hexahedral Finite Element Meshes,” International Journal for Numerical Methods in Engineering, Vol.39, pp.3327-3349 (1996).
[33] Folwell, Nathan T. and Scott A. Mitchell, ”Reliable Whisker Weaving via Curve Contraction,” Proceedings 7th International Meshing Roundtable, pp.365-378 (1998).
[34] Matthias Müller-Hannemann, “Hexahedral Mesh Generation by Successive Dual Cycle Elimination,” Proceedings 7th International Meshing Roundtable, pp.379-393 (1998).
[35] Matthias Müller-Hannemann, “Shelling Hexahedral Complexes for Mesh Generation,” Journal of Graph Algorithms and Applications, Vol.5 (5), pp.59-91 (2001).
[36] Borden, Michael J., Steven E. Benzley, and Jason F. Shepherd, “Hexahedral Sheet Extraction,” Proceedings 11th International Meshing Roundtable, pp.147-152 (2002).
[37] Min, Weidong, “Generating Hexahedron-Dominant Mesh Based on
Shrinking-Mapping Method,” Proceedings 6th International Meshing Roundtable, pp.171-182 (1997).
[38] Meyers, Ray J., Timothy J. Tautges, and Philip M. Tuchinsky, “The Hex-Tet Hex-Dominant Meshing Algorithm as Implemented in CUBIT,” Proceedings 7th International Meshing Roundtable, pp.151-158 (1998).
[39] Mitchell, Scott A., “The All-Hex Geode-Template for Conforming a Diced Tetrahedral Mesh to any Diced Hexahedral Mesh,” Proceedings 7th International Meshing Roundtable, pp.295-305 (1998).
[40] Leland, Robert W., Darryl J. Melander, Ray W. Meyers, Scott A. Mitchell, and Timothy J. Tautges, “The Geode Algorithm: Combining Hex/Tet Plastering, Dicing and Transition Elements for Automatic, All-Hex Mesh Generation,” Proceedings 7th International Meshing Roundtable, pp.515-521 (1998).
[41] Yamakawa, Soji and Kenji Shimada, “Hexhoop: Modular Templates For Converting A Hex-Dominant Mesh To An All-Hex Mesh,” Proceedings 10th International Meshing Roundtable, pp.235-246 (2001).
[42] Li, T. S., R. M. McKeag, and C. G. Armstrong, “Hexahedral Meshing Using Midpoint Subdivision and Integer Programming,” Computer Methods in Applied Mechanics and Engineering, Vol.124, pp.171-193 (1995).
[43] Price, Mark, Clive Stops, and Geoffrey Butlin, “A Medial Object Toolkit for Meshing and Other Applications,” Proceedings 4th International Meshing Roundtable, pp.219-229 (1995).
[44] Armstrong, C. G., D. J. Robinson, R. M. McKeag, T. S. Li, S. J. Bridgett, R. J. Donaghy and C. A. McGleenan, “Medials for Meshing and More,” Proceedings 4th International Meshing Roundtable, pp.277-288 (1995).
[45] Price, M. A. and C. G. Armstrong, “Hexahedral Mesh Generation by Medial Surface Subdivision: Part I,” International Journal for Numerical Methods in Engineering, Vol.38, pp.3335-3359 (1995).
[46] Price, M. A. and C. G. Armstrong, “Hexahedral Mesh Generation by Medial Surface Subdivision: Part II,” International Journal for Numerical Methods in Engineering, Vol.40, pp.111-136 (1997).
[47] Sheffer, A., M. Etzion, A. Rappoport, and M. Bercovier, “Hexahedral Mesh Generation using the Embedded Voronoi Graph,” Proceedings 7th International Meshing Roundtable, pp.347-364 (1998).
[48] Peter Sampl, “Semi-Structured Mesh Generation Based on Medial Axis,” Proceedings 9th International Meshing Roundtable, pp.21-32 (2000).
[49] Quadros, W. R. and Kenji Shimada, “Hex-Layer: Layered All-Hex Mesh Generation on Thin Section Solids via Chordal Surface Transformation,” Proceedings 11th International Meshing Roundtable, pp.169-182 (2002).
[50] Sheehy, D. J., C. G. Armstrong, and D. J. Robinson, “Computing the Medial Surface of a Solid from a Domain Delaunay Triangulation,” Symposium on Solid Modeling and Applications, pp.201-212 (1995).
[51] Reddy, J. M. and G. M. Turkiyyah, “Computation of 3D Skeletons Using a Generalized Delaunay Triangulation Technique,” Computer-Aided Design, Vol.27, No.9, pp. 677-694 (1995).
[52] Ramamurthy, R. and R. T. Farouki, “Voronoi Diagram and Medial Axis Algorithm for Planar Domains with Curved Boundaries I. Theoretical Foundations,” Journal of Computational and Applied Mathematics, pp. 119-141 (1999).
[53] Ramamurthy, R. and R. T. Farouki, “Voronoi Diagram and Medial Axis Algorithm for Planar Domains with Curved Boundaries II. Detailed Algorithm Description,” Journal of Computational and Applied Mathematics, pp. 253-277 (1999).
[54] Fabbri, R., L. F. Estrozi, and L. da F. Costa, “On Voronoi Diagram and Medial Axes,” Journal of Mathematical Imaging and Vision Archive, Vol.17-1, pp. 27-40 (2002).
[55] Culver, T., John Keyser, and D. Manocha, “Exact Computation of the Medial Axis of a Polyhedron,” Source Computer Aided Geometric Design Archive, Vol.21-1, pp. 65-98 (2003).
[56] Sherbrooke E. C., N. M. Patrikalakis, Franz-Erich Wolter, “Differential and Topological Properties of Medial Axis Transforms,” Graphical Models and Image Processing, Vol.58-6, pp. 574-592 (1996).
[57] Felkel, P. and S. Obdrzalek, “Straight Skeleton Implementation,” Proc. of Spring Conference on Computer Graphics, pp. 210-218 (1998).
[58] Joan-Arinyo, R., L. Perez, and J. Vilaplana, “Computer the Medial Axis Transform of Polygonal Domains by Tracing Paths,” Computer Graphics, (1999).
[59] Ang, Pin Yang and Cecil G. Armstrong, “Adaptive Curvature-Sensitive Meshing of the Medial Axis,” Proceedings 10th International Meshing Roundtable, pp. 155-165 (2001).
[60] Owen, S. J., “A Survey of Unstructured Mesh Generation Technology,” Proceedings 7th International Meshing Roundtable, pp. 239-267 (1998).
[61] Guo, We-Ker, ”3-D Model Reconstruction and Pre-Processing Software Development for Finite Element Analysis,” Master thesis, Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan (2003).
[62] Blacker, T., “Automated Conformal Hexahedral Meshing Constrains, Challenges and Opportunities,” Engineering with Computers, Vol.17, pp. 201-210 (2001).
[63] Owen, S. J., “Non-Simplical Unstructured Mesh Generation,” Ph. D. Dissertation, Department of Civil and Environmental Engineering, Carnegie Mellon University, (1999).
[64] Blum, H., “A Transformation for Extracting New Descriptions of Shape,” Models for the perception of speech and visual form, M.I.T. Press, pp. 362-380 (1967).